Speaking to the BBC after telling the Labour Party's annual conference that Brexit had "done more harm than good", Burnham said Britain should examine different models for its long-term relationship with Europe and decide which offered the best path forward.
"We could stay as we are. That's definitely an option, if people think this is the right place to stay," Burnham told the broadcaster.
"We could look at what George Osborne has said about a customs union. We could look at what the Liberal Democrats said at their conference about single market, or we could go all the way."
Osborne, a former Conservative finance minister, has argued that Britain should pursue a customs union with the EU, while the Liberal Democrats have called for Britain to rejoin the bloc's single market.
Burnham said the options should be assessed based on "what is doable" and the advantages and disadvantages of each.
Britain voted to leave the European Union in 2016, ending more than four decades of membership and triggering years of political upheaval, with successive governments grappling with the economic and diplomatic consequences of that vote.
Earlier on Tuesday, Burnham said he would use a November UK-EU summit to outline options for Britain's long-term relationship with the bloc.
Burnham, who replaced Keir Starmer in July, has previously said he would like Britain to rejoin the European Union in his lifetime. He has argued that Britain could not afford another decade of uncertainty over the issue.
"I am very clear, and I will say this bluntly: Brexit has caused more harm than good," he said.
"So the question is, what do we do about that?"
